Discrete-time low-gain control of uncertain infinite-dimensional systems

Using a frequency-domain analysis, it is shown that the application of a feedback controller of the form k/(z-1) or kz/(z-1), where k∈R, to a power-stable infinite-dimensional discrete-time system with square transfer-function matrix G(z) will result in a power-stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that i) all the eigenvalues of G(1) have positive real parts, and ii) the gain parameter k is positive and sufficiently small, Moreover, if G(1) is positive definite, we show how the gain parameter gain X can be tuned adaptively. The resulting adaptive tracking controllers are universal in the sense that they apply to any power-stable system with G(1)>0; in particular, they are not based on system identification or plant parameter estimation algorithms, nor is the injection of probing signals required. Finally, we apply these discrete-time results to obtain adaptive sampled-data low-gain controllers for the class of regular systems, a rather general class of infinite-dimensional continuous-time systems for which convenient representations are known to exist, both in state space and in frequency domain. We emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval